Adaptive Synchronization of the Fractional-Order LÜ Hyperchaotic System with Uncertain Parameters and Its Circuit Simulation

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1 9 Journal of Uncertain Systems Vol.6, No., pp.-9, Online at: Adaptive Synchronization of the Fractional-Order LÜ Hyperchaotic System with Uncertain Parameters and Its Circuit Simulation Sheng Miao,*, Shi-un Hua,, Wen-feng Hao, Faculty of Science, Jiangsu University, Zhenjiang, 3, Jiangsu Province, P.R. China Chinese Machinery Industry Key Laboratory for Mechanical Structure Damage Detection Technology, Jiangsu University Zhenjiang, 3, Jiangsu Province, P.R. China Received 3 December 9; Revised 5 September Abstract In this article, the adaptive control scheme with only two controllers is applied to synchronization of the fractionalorder hyperchaotic Lü system with unnown parameters. Based on fractional stability theory of fractional-order systems, adaptive controllers and the parameter updating rule are designed. Numerical simulations confirm the effectiveness of the proposed synchronization approaches. Especially, the circuit experiment simulations also demonstrate that the experimental results are in agreement with numerical simulations. World Academic Press, UK. All rights reserved. Keywords: fractional order hyperchaotic system, synchronization, adaptive controllers, circuit simulation Introduction The fractional calculus dates from 7th century, but its applications to physics, engineering and control processing are just a recent subject of interest [, 7]. Many systems exist in interdisciplinary fields, such as dielectric polarization [5], viscoelastic systems [8], uantitative finance [] and uantum evolution of complex systems [9]. It is nown that some fractional systems behave chaotically, for example, the fractional-order Chua s circuit [6], the fractionalorder unified chaotic system [7], the fractional-order Rössler system and so forth []. On the other hand, synchronization of chaotic fractional-order system has recently received considerable attention owing to its potential applications in secure communication and control processing, for example, Zhang et al. adopted Pecora-Carroll method, the linear feedbac control and the bidirectional coupling to realize chaotic synchronization of the Ruclidge system [9]; Wang et al. analysis the synchronization conditions of the fractional order chaotic systems with activation feedbac method [6]; Lu proposed to realize the synchronization of the fractional order unified chaotic systems using state observer in Ref.[]; Wu investigate the synchronization of fractional-order Chen hyperchaotic systems based on Laplace transform theory [8].To our best nowledge, we can find that some methods such as linearization feedbac control method and active control scheme, eliminate nonlinear terms of systems when designing controllers, which mae the coefficient matrix of the system to be the constant matrix. Although this scheme can control the fractional-order chaotic system to synchronize, it costs too much. The adaptive control method proposed in this paper can achieve synchronization of fractional-order hyperchaotic systems only using two controllers, and we can see the feasibility of this techniue. Moreover, since hyperchaotic systems have two positive Lyapunov exponents at least. The trajectories of hyperchaotic systems show more complex dynamical behaviors because they extend towards many directions. Complex hypechaotic signal can increase the reliability of encryption techniue in chaotic secure communication. Therefore, study on synchronization of the fractional-order hyperchaotic systems will be a crutial uestion for discussion in aspects of chaos application. In the present paper, we propose the adaptive synchronization method with only two controllers for the fractional-order hyperchaotic Lü systems with unnown parameters. This scheme, based on stability theory of * Corresponding author. danny984@63.com (S. Miao).

2 S. Miao et al.: Adaptive Synchronization of the Fractional-Order LÜ Hyperchaotic System fractional-order systems is simple, theoretically rigorous and convenient to realize synchronization. What s more, not only do numerical simulations be performed to verify the effectiveness of the proposed synchronization techniues, but circuit experiment simulation results show the effectiveness of the presented method. Fractional Calculus Predictor-Corrector Algorithm There are several definitions of a fractional differential system. The most-used one in engineering field may be the Riemann-Liouville fractional derivatives, defined by m ( m) D x() t J x (), t () where m [ ], i.e., m is the first integer which is not less than. J ( ) is the -order Riemann-Liouville integral operator with expression: t J yt () ( t ) y( ) d ( ). () Here stands for the Gamma function. The operator D is generally called -order Caputo differential operator [3]. According to Refs.[4-5], the predictor-corrector method, namely the generalized Adams-Bashforth-Moulton method is described. Consider the following fractal order differential euations: where This is euivalent to the following Volterra integral euations: D y() t f(, t y()) t, t T, ( ) ( ) and y () y,,,, m. (3) m ( ) t t yt () y ( ts) f(, sys ( )) ds.! ( ) Set h T N, t nh, n,,, NZ. Then, one can discrete this euation as n (4) t h h y ( t ) y f( t, y ( t )) a f( t, y ( t )) m n ( ) n p h n n h n j, n j h j! ( ) ( ) j (5) n ( na)( n), j ajn, ( n j) ( n j) ( n j), j n, j n, t h y ( t ) y b f( t, y ( t )) m n p ( ) n h n j, n j h j! ( ) j, h b jn, (( n j) ( n j) ). The algorithm convergence order of an error series in the above numerical scheme is p, that is p max j,,, N yt ( j) yh( tj) Oh ( ), where p min(, ).Numerical solution of a fractional-order system can be determined by applying the mentioned method. 3 Synchronization via Adaptive Control Recently, Min et al. proposed the fractional-order Lü hyperchaotic systems in Ref. [4] which is described by

3 Journal of Uncertain Systems, Vol.6, No., pp.-9, 3 d x ay ( x) w d y cy xz (6) d z xy bz d w xz dw where is the fractional-order, when =.95, (a, b, c, d) = (36, 3,, ), the system behaves hyperchaotic. Fig. exhibits hyperchatioc attractor. 3. Design of Controllers Figure : Hyperchaotic attractors of the fractional-order Lü system with the order =.95 In this section, we introduce adaptive control method synchronize two identical hyperchaotic fractional-order Lü system. Here, only two controllers are adopted. The drive system is described by d x ay ( x) w d y cy xz (7) d z xy bz d w xz dw whereas the response system is d x a( y x) w d y cy xz u (8) d z xy bz d w x z dw u where a, b, c, d are unnown parameters, which need to be estimated in the response system, u=[u, u ] T are controllers which are to be designed. Now, we define the state errors between system (7) and (8) as e = x - x, e = y - y, e 3 = z - z, e 4 = w - w. Subtraction the system (7) from system (8), the error dynamical system can be expressed by

4 4 S. Miao et al.: Adaptive Synchronization of the Fractional-Order LÜ Hyperchaotic System de a( e e) ea ( y x) e 4 de ce e c y ez ex 3 u (9) de3 ey ex be 3 ez b de4de 4ed wze xe 3u where e a = a - a, e b =b - b, e c = c - c, e d = d - d. Our objective is to design effective controllers u and the update law of parameters to mae the response system (8) and drive system (7) to achieve synchronization. We choose the controllers () and the update law of parameters () as follows: u e e ee () u e4 e4 e e4 * * * where e, e, i (i=,) are real constants. And the update law of parameters: d e,( ) d e4,( ) dea ( y x) e () deb ze 3 dec ye deb we 4. Theorem For any initial conditions, the drive systems (7) and response system (8) are globally asymptotically synchronized by the control law () and update law (). Proof: Combining the control law (), update law () and error system, we can get such that: d e d e d e 3 a a ( y x) e d e z c x y e 4 e y x b z e3 d e z a x d w e4 e4 ( y x) e a. d e b z eb y e c d e c w ed e e d e d e 4 e d e d e

5 Journal of Uncertain Systems, Vol.6, No., pp.-9, 5 Let a a ( y x) z c x y e y x b z z x d w e4 ( yx) A z y w e e4 and let the number is eigenvalue of matrix A, if there exists a nonzero vector (,,, ) T,which is an eigenvector of A associated with the eigenvalue, we can get A. () Let tae the conjugate transpose on both sides of euation above, we have H T H A. (3) H And we use multiply on the left of euation () and postmultiply in the euation (3), respectively, then mae an addition after the previous process. We also have H T H ( A A ) ( ). (4) Owing to the attractiveness of the attractor, there exists m, such that max( x, y, z, w ) m, max( x, y, z, w ) m, to our best nowledge,. Thus, i j j i i j j i a z y z a a z c H H T ( ) ( A A ) ( 3 4) y x 3 b 4 z x d a m m m a a m c ( 3 4 ) m m 3 b 4 m m d ( 3 4 ) P 3 4 m Hence, if we choose, 4ab ( a m) b, c 4ab m

6 6 S. Miao et al.: Adaptive Synchronization of the Fractional-Order LÜ Hyperchaotic System ( c )[ m a m ( m) b ( m) ] m ( a m) /4, d ( c)(4 ab m ) b( a m) and we can get that P is positive definite matrix. Therefore ( ) H ( 3 4 ) P 3 4 Obviously,, and any eigenvalue of matrix A is satisfying arg( i ).5, based on the stability theory of the fractional-order systems [3], the error e i (i=,,3,4) will converge to zero as t which implies that the system (7) and (8) are globally synchronized. 3. Numerical Simulation In the numerical simulation with the predictor-corrector method, select the true values of unnown parameters of the drive system as a=36, b=3, c=, and d=, and tae the initial estimated parameters as a ()=-5, b ()=-5/3, c ()=-8, and d ()=-.9; we choose, ()= ()=. Also, the initial condition x ()=3, y ()=-4, z ()=, w ()=, x ()=-3, y ()=4, z ()=-, w ()=-. From Fig..(a), it can be seen that the synchronization error converges to zero. System (7) and (8) have achieved absolute synchronization. In Fig. (b) and (c), the estimates a, b, c, d of the unnown parameters converge to a =a=36, b =b=3, c =c=, d =d=, gradually with time increase, and, will converge to constant, respectively, as t..

7 Journal of Uncertain Systems, Vol.6, No., pp.-9, 7 Figure : The synchronization errors between system (7) and (8) and the estimate value of parameters a, b, c, d, and with time t 3.3 Circuit Experimental Verification As is well nown, using the standard integer order operators to approximate the fractional operators is an effective scheme to solve this ind of problem. In Ref.[], approximations for / s with =.-.9 in step size. were given with errors of approximately db. Here, we tae =.95 as fractional order of hyperchaotic system. The following transfer function approximation method presented in Ref. []:.8s 8.64s s s s.7574s.3 and the corresponding circuit unit is displayed in Fig.3. Thus, the transfer function H(s) between A and B can be expressed as follows: C C C C C C R R R ( ) s C C C R R R3 RRR 3 s C C C3 CC CC3 CC 3 H() s. (5) C ( s/ RC )( s/ R C )( s/ R C ) where C is the unit parameter. Letting C = F and F(s)=H(s)C = / s and comparing E.(5) with Expression of.95 / s, we can obtain the values of resistances and capacitances as follows:r =77.4, R =54, R 3 =5, C =.678 μf, C = μf, C 3 =3.643 μf. AD633 is used as multiplier with an output coefficient of., LM74 chip is an operational amplifier. Other values of resistances in circuit are indicated in Fig.4. For simplicity, we choose the parameters of the drive system and response system with R a = R a =.78, R b = R b =33.3, R c = R c =5, R d = R d =. The parts in the dashed line are the circuitry of controller u, R, R can be adjustable. The circuit simulation results are depicted in Fig.5. From it, we can find that the slope of state variables euals. It is clear that the drive system (7) and response system (8) achieve synchronization. Figure 3: Circuit diagram of fractional-order /s.95

8 8 S. Miao et al.: Adaptive Synchronization of the Fractional-Order LÜ Hyperchaotic System Controllers U Figure 4: Circuit of synchronization between system (7) and (8) a. x - x (V/div) b. y - y (V/div) c. z - z (V/div) d. w - w (5V/div) Figure 5: Circuit experiment of states synchronization between system (7) and (8)

9 Journal of Uncertain Systems, Vol.6, No., pp.-9, 9 4 Conclusion In this paper, we have studied the synchronization of the fractional-order Lü hyperchaotic systems applying adaptive approaches based on the stability theory of the fractional-order system. The synchronization techniues are simple, theoretically rigorous and convenient to realize synchronization. Numerical simulations are given to verify the effectiveness of the proposed synchronization schemes, and also, circuit experiment simulation results are in agreement with numerical simulations. Also, this method can also be extended to other hyperchaotic fractional-order differential systems. Acnowledgements This research is supported by the National Natural Science Foundation of China (Grant No.979). References [] Ahmad, W.M., and J.C. Sprott, Chaos in fractional-order autonomous nonlinear systems, Chaos, Solitons and Fractals, vol.6, pp , 3. [] Butzer, P.L., and U. Westphal, An Introduction to Fractional Calculus, World Scientific, Singapore,. [3] Caputo, M., Linear models of dissipation whose Q is almost freuency independetnt-Ⅱ, Geophysical Journal of the Royal Astronomical Society, vol.3, pp , 967. [4] Diethelm, K., and N.J. Ford, Analysis of fractional differential euations, Journal of Mathematical Analysis and Applications, vol.65, pp.9 48,. [5] Diethelm, K., N.J. Ford, and A.D. Freed, A predictor-corrector approach for the numerical solution of fractional differential euations, Nonlinear Dynamics, vol.9, pp.3,. [6] Hartley, T.T., C.F. Lorenzo, and H.K. Qammer, Chaos on a fractional Chua s system, IEEE Transactions on Circuits and Systems- I, Fundamental Theory and Applications, vol.4, pp , 995. [7] Hilfer, R., Applications of Fractional Calculus in Physics, World Scientific, New Jersey,. [8] Koeller, R.C., Application of fractional calculus to theory of viscoelasticity, Journal of Applied Mechanics, vol.5, pp.99 37, 984. [9] Kusnezov, D., A. Bulgac, and G.D. Dang, Quantum levy processes and fractional inetics, Physical Review Letters, vol.8, pp.36 39, 999. [] Lasin, N., Fractional maret dynamics, Physica A, vol.87, pp.48 49,. [] Li, C.G., and G. Chen, Chaos and hyperchaos in the fractional-order Rössler euations, Physica A, vol.34, pp.55 6, 4. [] Lu, J., C. Liu, Z. Zhang, and X. Chen, State-observer based synchronization control between fractional-order unified chaotic systems, Journal of Xi an Jiaotong University, vol.4, pp.497 5, 7. [3] Matignon, D., Stability results for fractional differential euations with applications to control processing, IMACS, IEEE- SMC, vol., pp , 996. [4] Min, F.H., Y. Yu, and C.J. Ge, Circuit implementation and tracing control of the fractional-order hyper-chaotic Lü system, Acta Physica Sinica, vol.58, pp , 9. [5] Sun, H.H., A.A. Abdelwanhab, and B.Onaral, Linear approximation of transfer function with a pole of fractional power, IEEE Transactions on Automatic Control, vol.9, pp , 984. [6] Wang, X.Y., and J.M. Song, Synchronization of the fractional order hyperchaotic Lorenz systems with activation feedbac control, Communications in Nonlinear Science and Numerical Simulation, vol.4, pp , 9. [7] Wang, J.W., and Y.B. Zhang, Designing synchronization schemes for chaotic fractional-order unified systems, Chaos, Solitons and Fractals, vol.3, pp.65 7, 6. [8] Wu, X., and Y. Lu, Generalized projective synchronization of the fractional-order Chen hyperchaotic system, Nonlinear Dynmics, vol.57, pp.5 35, 9. [9] Zhang, Y., and T. Zhou, Three schemes to synchronize chaotic fracional-order ruclidge systems, International Journal of Modern Physics B, vol., pp.33 44, 7.